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The only way to produce antiprotons would be to collide some particles together and then filter out just the antiprotons (as there are no stable particles that we know of containing antiprotons).

Fermilab produces the protons and antiprotons in a couple of stages. It starts with negatively ionised hydrogen and accelerates it using a Cockcroft-Walton accelerator to 750keV, by basically cycling the ions through a potential difference multiple times. These ions are then fed into a linear accelerator that uses an oscillating electroc field to accelerate them to 400 MeV. The electrons are then stripped from the ions using a carbon foil, ending up with 400MeV protons. These then go into the Booster which accelerates them to 8GeV.

This is the proton production stage and following this, the 8GeV protons are directed to the first ring of the Tevatron (the Main Injector) where they can be accelerated up to 150GeV. Once the protons reach 120GeV, some are collided with a nickel target, producing a shower of particles. The antiprotons are filtered and directed back to the Main Injector, which accelerates both the protons and antiprotons to 150GeV. The two beams are now fed into the main accelerator (the Tevatron) which accelerates them to 1TeV and crosses them at the detector site, producing the shower of particles that is detected and analysed :)

It's worth noting that all beams of unstable particles (muons, pions, neutrons, etc...) are produced colliding a beam of stable stuff with a target and selecting the desired products from the resulting spray. (We'll treat anti-protons as "unstable" for these purposes 'cause you can't easily hold them...)
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dmckee♦Jan 15 '11 at 19:16